We don't necessarily need continent-sized high-speed rail. Our population density isn't nearly as low as you think it is, or as the official numbers suggest. The problem with the official numbers is that population density is not uniform: something like 2/3 of America's population lives east of the Mississippi River. And much of that lives in the northeast corridor. When you add in all the vast, unoccupied areas in the western states like Wyoming, then the overall density seems low.
So, instead of worrying about continent-sized high-speed rail networks, maybe we should start by thinking about building a high-speed rail network that spans the Eastern Seaboard states, where so much of the population is packed, then then expand out from there (Chicago, Memphis, New Orleans, Houston, etc.). We've already tried this with Amtrak and their Acela Express, but it's kinda lame: expensive and not that fast.
If we just built a seriously fast high-speed train between Boston and DC (stopping at NYC, Philly, and Baltimore along the way), and priced it competitively with air travel, then we'd be getting somewhere. Extend it down to RTP, Atlanta, and Florida next, and then a western spur to Cleveland and Chicago.
A separate high-speed rail network along the west coast, linking San Diego, LA, SanFran, Portland, Seattle, and Vancouver would also make a lot of sense.
What probably wouldn't make so much sense is trying to link the east and west coasts with such a system. Rail (even HSR) is just too slow to compete with air travel for distances that far.
Of course, Musk's Hyperloop idea might make more sense than any of this, though it'd basically be serving the same purpose in mostly the same way, just with tubes instead of rails. But its speeds would be even higher than air travel, so it could really replace air travel for major corridors, leaving air travel for places where Hyperloop isn't built yet, or for any travel which crosses large bodies of water.